Harnack inequality for degenerate balanced random walks

We consider an i.i.d. balanced environment \(\omega(x,e)=\omega(x,-e)\), genuinely d dimensional on the lattice and show that there exist a positive constant \(C\) and a random radius \(R(\omega)\) with streched exponential tail such that every non negative \(\omega\) harmonic function \(u\) on the ball \(B_{2r}\) of radius \(2r>R(\omega)\), we have \(\max_{B_r} u \leq C \min_{B_r} u\). Our proof relies on a quantitative quenched invariance principle for the corresponding random walk in balanced random environment and a careful analysis of the directed percolation cluster. This result extends Martin Barlow's Harnack's inequality for i.i.d. bond percolation to the directed case. This is joint work with N. Berger, M. Cohen, and X. Guo.